Proceedings of the 2005 IEEE Conference on Control Applications Toronto, Canada, August 28-31, 2005
TA3.3
Adaptive Backstepping for Nonlinear Switching Control AMB System with Base Excitation Selim Sivrioglu and Muneharu Saigo Abstract— This research aims to test the limits of control effectiveness for an active magnetic bearing (AMB) system subjected to external acceleration disturbances. An adaptive output backstepping controller is designed to compute nonlinear control currents of the magnetic bearing by accepting that the rotor and AMB parameters are unknown. The backstepping controller is experimentally verified and the results are compared with PID control results for the maximum amplitude of applicable disturbance. Index Terms—Adaptive output backstepping, zero-bias active magnetic bearing, flywheel system, nonlinear control.
I. Introduction Development of high efficient and clean energy storage systems become an important research topic due to environmental problems. Flywheel energy storage systems which store rotating kinetic energy by high-speed rotation have a promising future because of their cleanliness and high power energy storage [1]. Active magnetic bearings have potential to be utilized in energy storage flywheel systems due to noncontact levitation of the rotor. An AMB provides contact-free suspension of a rotor, thus, friction is not present and lubrication is not needed. Therefore, the lifetime of an active magnetic bearing outperforms that of a mechanical bearing. These characteristics make active magnetic bearings attractive for many applications especially flywheel energy storage systems. Despite such advantages, the stiffness of a magnetic bearing is low compared with a mechanical bearing. Since flywheels are rotating parts with large kinetic energy capacity depending on the size, they have destructive potential and may cause deadly damages if some failure occurs at high speed spinning. For large scale flywheel applications, earthquake like disturbances may cause some failures in control systems. Therefore, the designed controller should be tested for such kind of disturbances. Nonlinear control of the active magnetic bearings has been studied using nonlinear switching techniques in [2][4]. Also, integral backstepping control design has been applied to an AMB system in reference [5]. In this study, an adaptive output feedback type backstepping control is presented to achieve better performance in the case of acceleration disturbances.
vibrating base which is connected to a shaker for acceleration disturbance. The flywheel system uses in general three active magnetic bearings to suspend the rotor, i.e. two magnetic actuator in radial direction and one in axial direction. However, the axial bearing is not considered in the following control design unless otherwise stated. Figure 2 illustrates the flywheel-rotor and placement of the radial magnetic actuators in upper and lower locations in xGz plane. The parameters of the flywheel AMB system is given in Table I. Note that four other magnetic actuators are also placed in yGz plane symmetrically, but are not shown in Fig.2. The objective of the control is to provide the stability of the AMB system under the applicable maximum amplitude of acceleration disturbances.
Fig. 1: Photo of the AMB system
f3
f1 Tx
l
T y
II. Problem Statement Consider the vertically designed AMB system depicted in Fig.1. In this setup, the AMB system is placed on a
f7
S. Sivrioglu is with the Faculty of Engineering, Gebze Institute of Technology, Gebze-Turkey. e-mail:
[email protected] M. Saigo is with the Advanced Manufacturing Research Institute, AIST, Tsukuba-shi, Japan.
0-7803-9354-6/05/$20.00 ©2005 IEEE
f5
Fig. 2: Flywheel rotor-AMB system
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x
TABLE 1: Parameters of the Symbol Value M 10.78 Ir 0.150573 Ia 0.0275912 lu 0.22467 0.13033 ll l1 0.26817 0.17409 l2 K 2.463 × 10−7 X0 , Y0 0.5 × 10−3 10000 Ks
AMB system Unit kg kgm2 kgm2 m m m m N m2 /A2 m V/m
IV. ADAPTIVE OUTPUT BACKSTEPPING A. Nonlinear Observer In practice, only the displacement of the rotor is measured, and the velocity is not available for the feedback. To estimate the unmeasured state x2 , an exponentially convergent observer is introduced into the control system such as x ˆ2 = η + θλ + kx1
III. NONLINEAR SWITCHING CONTROL AMB The nonlinear control defined here serves to switch the control current of the electromagnets of the active magnetic bearing according to the rotor position. Unlike the linear control, no bias current is employed in the proposed nonlinear control. In an attractive type bearing configuration of a pair magnet, when the rotor approaches one of the magnets, the coil current in that magnet switches to zero while the coil current in the magnet on the opposite side is switched on to generate attractive force. The switching process continues until the rotor is brought to the origin and stabilized. The mathematical model of the flywheel AMB system is derived and the upper and lower AMB equations are decentralized (see Appendix). After this process, each axis of the AMB becomes independent of each other. The structure of the nonlinear switching control may be defined only for one axis. Using the variable transformations x1 = xu and x2 = x˙ u in Equation (28), the second order system is obtained as x˙ 1 = x2 u1 x˙ 2 = θβ1 (x1 ) −θβ3 (x1 ) u3 y = x1
θ = au K u 1 1 β1 (x1 ) = , β3 (x1 ) = (X0 − x1 )2 (X0 + x1 )2
(2)
Note that the nonlinear functions β1 and β3 are strictly positive functions of the state x1 . It seems that Equation (1) has multiple inputs but in reality only one control input is effective at any time depending on the rotor position such as x˙ 1 = x2 x˙ 2 = θβu y = x1
where η and λ denote the states of the filters. Also, k is a positive parameter. The first filter is for the part of the plant that does not contain the unknown parameter θ and the second one is for the unknown part of the plant. The filters are defined as η˙ = −kη − k 2 x1 λ˙ = −kλ + β(x1 )u
= x2 − x ˆ2 ˙ = x˙ 2 − η˙ − θλ˙ − k x˙ 1 = k(η + θλ + kx1 ) − kx2 = −k
u = u3 ,
u1 = 0,
β = −β3 (x1 )
if x1 < 0
u = u1 ,
u3 = 0,
β = β1 (x1 )
(3)
(6)
B. Control Design The controller will be designed as defined in Reference [6]. The objective of the control is to track the reference yr (t) with the system output y(t). The tracking error is defined as z1 = y − yr (7) The derivative of z1 is obtained as z˙1 = x˙ 1 − y˙ r = x2 − y˙ r
(8)
Since x2 is not measured, the estimate of it will be used. Now substituting Equation (4) into (8), the derivative of z1 becomes z˙1 = η + θλ + kx1 + (9) Note that the reference input yr is zero for the rotor magnetic bearing system. In Equation (9), the only variable that contains control input u is λ. Thus λ may be used to define the second error variable such as z2 = λ − α1
(10)
where α1 is a stabilizing function and is chosen as ¯1 α1 = ρˆ(−c1 z1 − d1 z1 − kx1 − η) = ρˆα
if x1 ≥ 0
(5)
If the initial conditions are η(0) = 0 and λ(0) = 0, then it is guaranteed that the estimation error exponentially converges to zero.
(1)
where u1 = i21 and u3 = i23 are the control inputs. y denotes the output of the system. The unknown parameter θ and the nonlinear functions β1 and β3 are defined as
(4)
(11)
where c1 > 0, d1 > 0. Note that to eliminate λ in Equation (9), the stabilizing function α1 is multiplied by ρˆ which is an estimate of the parameter ρ = 1/θ. Moreover, unlike in the integrator backstepping procedure, d1 is added to
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counteract the estimation error . Now, the derivative of z1 becomes z˙1 = η + θ(z2 + α1 ) + kx1 + α1 ] + kx1 + = η + θ[z2 + (ρ − ρ˜)¯ = θz2 − c1 z1 − d1 z1 − θρ˜α ¯1 +
(12)
where ρˆ = ρ − ρ˜. Here ρ˜ denotes the error in the estimation of ρˆ. A candidate Lyapunov function for the first error variable is defined as 1 1 1 2 θ(ρ − ρˆ)2 + V1 = z12 + (13) 2 2γ 2kd1 where γ is the adaptation gain. The derivative of V1 is obtained as 1 1 ˙ V˙ 1 = z1 z˙1 − θ(ρ − ρˆ)ρˆ˙ + γ kd1 1 α1 z1 + ρˆ˙ ) = θz1 z2 − c1 z12 − θ(ρ − ρˆ)(¯ γ 2 1 1 2 1 2 − d1 z1 − + − 2d1 4d1 d1
where θˆ is the estimate of the unknown parameter θ. Also, θ˜ represents error in this estimation. From Equation (19), the control input u is chosen as 2 ∂α1 1 ˆ 1 + kλ z2 − θz −c2 z2 − d2 u= β(x1 ) ∂y ∂α1 ˆ + kx1 ) + ∂α1 (−kη − k 2 x1 ) (20) (η + θλ + ∂y ∂η ∂α1 ˙ + ρˆ ∂ ρˆ Substituting (20) into (18), the derivative of z2 becomes
2 ∂α1 ˜ 1 z˙2 = −c2 z2 − d2 z2 − (θ − θ)z ∂y ∂α1 ∂α1 ˜ θλ − − ∂y ∂y
The Lyapunov function is augmented for the second error variable z2 as follows 1 1 ˆ 2 + 1 2 (θ − θ) V2 = V1 + z22 + 2 2γ 2kd2
−d1 z12 +z1
3 2 1 ≤ θz1 z2 − c1 z12 − θ(ρ − ρˆ)(¯ α1 z1 + ρˆ˙ ) − γ 4d1
(14) The θ(ρ− ρˆ) term in the above inequality can be eliminated using the update law as follows: ρˆ˙ = −γ α ¯ 1 z1
Here, the stabilizing function α1 is a function of y, η and ρˆ. The derivative of α1 is obtained as α˙ 1 =
∂α1 ∂α1 ˙ ∂α1 y˙ + η˙ + ρˆ ∂y ∂η ∂ ρˆ
(17)
Substituting Equations (5) and (17) into Equation (16), the derivative of z2 becomes ∂α1 (η + θλ + kx1 + ) ∂y ∂α1 ∂α1 ˙ (−kη + k 2 x1 ) − ρˆ − ∂η ∂ ρˆ
z˙2 = −kλ + β(x1 )u −
(18)
Equation (18) is not a desired form because the unknown parameter θ appears. Moreover, the disturbance is mul1 tiplied by the nonlinear term ∂α ∂y . To employ nonlinear damping and to eliminate the unknown parameter, Equation (18) is equalized as follows: 2 ˜ ˆ 1 − d2 ∂α1 z2 − ∂α1 − ∂α1 θλ −c2 z2 − θz ∂y ∂y ∂y ∂α1 (19) (η + θλ + kx1 + ) = −kλ + β(x1 )u − ∂y ∂α1 ∂α1 ˙ − (−kη − k 2 x1 ) − ρˆ ∂η ∂ ρˆ
(22)
The derivative of V2 is derived as 1 ˆ θˆ˙ + 1 ˙ V˙ 2 = V˙ 1 + z2 z˙2 − (θ − θ) γ kd2 = −c1 z12 − c2 z22 2 ∂α1 1 1 2 1 2 3 2 z2 − − d2 + − − ∂y 2d2 4d2 d2 4d1
(15)
Since the term θz1 z2 remained in (14) , a global stability condition is not satisfied in this step. The second step is to expand the control design to include the error variable z2 . To this aim, the derivative of the second error variable is given as (16) z˙2 = λ˙ − α˙ 1
(21)
d2 (
∂α1 2 2 ∂α1 ∂y ) z2 − ∂y
z2
≤ −c1 z12 − c2 z22
3 ∂α1 1˙ 3 λz2 − θˆ − + 2 ∂y γ 4d1 4d2 (23) ˆ in where c2 > 0, d2 > 0. The unknown part of (θ − θ) Equation (23) is eliminated using the update law ∂α1 ˙ ˆ λz2 (24) θ = γ z1 z2 − ∂y ˆ z1 z2 − +(θ − θ)
It is clear that a global stability is maintained in the final Lyapunov function. The control currents i1 and i3 can be computed from the control input u obtained in Equation (20). V. Experiments A feedback control system is built using a high speed Texas Instrument digital signal processor(TMS320C6701) to realize experiments. The control system is a multi-input multi-output structure with four displacements measured by four eddy current position sensors and eight computed control current signals for actuators. The control inputs are supplied to electromagnets through D/A converters and power amplifiers. In rotor-AMB systems, trajectories of the geometric center point of the rotor and control currents are generally used to evaluate the control performance. Since the limits
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7
2 1
2 1
0 0
0.1
0.2 0.3 Time [ s ]
0.4
0 0
0.5
Displacement ( Z= 0 Hz )
0.25 [ mm ]
0.25 [ mm ]
0.5
Ŧ0.25
Ŧ0.25
Ŧ0.5 0
0.1
0.2 0.3 Time [ s ]
0.4
Ŧ0.5 0
0.5
0.25
0.25
u
yl Axis
0 Ŧ0.25 Ŧ0.5 Ŧ0.5
Ŧ0.25 0 0.25 x Axis [ mm ]
Ŧ0.5 Ŧ0.5
0.5
0.4
Control current ( Z= 0 Hz )
2
[A]
1
0 0
0.5
3
7
2
i
[A] i5
xl
3
0.1
Fig.3 Displacement of the rotor(Adaptive Control) Upper ( Z= 0 Hz )
0.2 0.3 Time [ s ]
0.4
0 0
0.5
Lower ( Z= 0 Hz )
Displacement ( Z= 50 Hz )
0.25
0.25
0.25
0.5
Ŧ0.25 Ŧ0.5 Ŧ0.5
Ŧ0.25 Ŧ0.25 0 0.25 xl Axis [ mm ]
0.4
0.5
Ŧ0.5 0
0.5
Fig.4 Orbit of the rotor(Adaptive Control)
0
yl
l
0
x
yl Axis
Ŧ0.25 0 0.25 xu Axis [ mm ]
0
[ mm ]
0.25
[ mm ]
0.5
[ mm ]
0.5
Ŧ0.5 Ŧ0.5
0.2 0.3 Time [ s ]
Displacement ( Z= 50 Hz )
0.5
Ŧ0.25
0.1
Fig.8 Control currents(PID Control)
0.5
0
0.5
Fig.7 Orbit of the rotor(PID Control) Control current ( Z= 0 Hz )
yl
0.2 0.3 Time [ s ]
Ŧ0.25 0 0.25 x Axis [ mm ] l
1 0.1
0.5
Ŧ0.25
u
Ŧ0.25
0.5
0.4
0
4
0.4
0.2 0.3 Time [ s ]
Lower Orbit ( Z= 0 Hz ) 0.5 [ mm ]
[ mm ]
Upper Orbit ( Z= 0 Hz ) 0.5
4
0.2 0.3 Time [ s ]
0.1
Fig.6 Displacement of the rotor(PID Control)
0.25
0.1
0.5
0
0.25
Ŧ0.5 0
0.4
yl
x
l
0
5
Ŧ0.5 0
0.2 0.3 Time [ s ]
Displacement ( Z= 0 Hz )
0.5
5
0
0.1
Fig.5 Control currents(Adaptive Control)
Displacement ( Z= 0 Hz )
Ŧ0.25
[ mm ]
3
i
3
[A]
4
[A]
4
i5
5
0.5
0
yu Axis
Control current ( Z= 0 Hz )
5
0.5
[ mm ]
[ mm ]
Displacement ( Z= 0 Hz )
Control current ( Z= 0 Hz )
y Axis
of the control effectiveness is aimed, the base is subjected to acceleration disturbance at a fixed frequency of 10 Hz but the amplitude of the disturbance is continuously increased up to to the touch down position of the rotor. Note that the AMB setup is placed on the base to make a 45 degree of angle with the axis of actuators so that the disturbance effect is equally distributed both x and y directions. The results are obtained for the maximum amplitude of the disturbance. In this study, the results of lower actuator location are presented because the disturbance effect is large compared to upper AMB location. Figures 3-4 show the displacement and orbits of the rotor for backstepping adaptive control case (ω = 0), respectively. The control currents of the lower actuator are presented in Fig.5. The control current characteristics reflect the proposed nonlinear switching control principle clearly. From Fig.6 to Fig.8, the PID control results are presented for the same location of the actuator. For comparison of both cases, the maximum applied acceleration disturbance is shown in Fig.15(a). As seen in this figure, the adaptive control maintains the stability of AMB system even the amplitude of disturbance approximately two times higher than PID control case. The control currents are almost same level for both case(see Fig.5 and Fig.8). Note that PID control has no switching dynamics as defined in Equation (3) and a 0.5 A bias current is applied to the magnetic actuators in PID control. The flywheel AMB system is rotated up to 60 Hz and acceleration disturbance is applied to the base at 50 Hz. From Fig.9 to Fig.14, the results are presented for both adaptive backstepping and PID control cases, respectively. For evaluation of the results, Fig.15(b) should be considered to distinguish the achieved control performance in adaptive control case.
Ŧ0.25
0.1
0.2 0.3 Time [ s ]
0.4
0.5
Ŧ0.5 0
0.1
0.2 0.3 Time [ s ]
0.4
Fig.9 Displacement of the rotor(Adaptive Control)
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0.5
0.1
Ŧ0.25
Amplitude
0
0
0.08
0.06
0.06
0.04
0.04
0.02 0 Ŧ0.02 Ŧ0.04 Ŧ0.06
Ŧ0.25
Ŧ0.25 0 0.25 xu Axis [ mm ]
Ŧ0.5 Ŧ0.5
0.5
Ŧ0.1 0
Ŧ0.25 0 0.25 xl Axis [ mm ]
4 [A]
4 3
3
7
2 1
2 1
0 0
0.1
0.2 0.3 Time [ s ]
0.4
0 0
0.5
0.1
0.2 0.3 Time [ s ]
0.4
0.5
Fig.11 Control currents(Adaptive Control) Displacement ( Z= 50 Hz )
0.25 [ mm ]
0.25 [ mm ]
0.5
Ŧ0.25
0.4
0
Ŧ0.5 0
0.5
0.25
[ mm ]
0.25 0 Ŧ0.25
0.1
0.2 0.3 Time [ s ]
0.4
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Ŧ0.25 Ŧ0.5 Ŧ0.5
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Ŧ0.25 0 0.25 xl Axis [ mm ]
0.5
Control current ( Z= 50 Hz )
4
4 [A]
5
3
7
2
i
i
5
[A]
Control current ( Z= 50 Hz )
1
3 2 1
0.2 0.3 Time [ s ]
0.4
PID control case 0.1
[s]
0.2
Time
fxu = (f1 − f3 ), fyu = (f2 − f4 ),
0
5
0.1
Ŧ0.1 0
0.3
0.4
0.5
[s]
0.5
0 0
0.1
0.2 0.3 Time [ s ]
Fig.14 Control currents(PID Control)
0.4
(25)
The control forces are
Fig.13 Orbit of the rotor(PID Control)
0 0
0.5
Mx ¨g = fxu + fxl Ir θ¨y = lu fxu − ll fxl M y¨g = fyu + fyl Ir θ¨x = −lu fyu + ll fyl
Lower Orbit ( Z= 50 Hz ) 0.5
yl Axis
yu Axis
[ mm ]
Upper Orbit ( Z= 50 Hz ) 0.5
Ŧ0.25 0 0.25 xu Axis [ mm ]
0.4
The equation of motion of the rigid rotor-active magnetic bearing system depicted in Fig.2 is derived as
Fig.12 Displacement of the rotor(PID Control)
Ŧ0.5 Ŧ0.5
0.3
Appendix
Ŧ0.25
0.2 0.3 Time [ s ]
0.2
The design of adaptive output backstepping control is presented for a nonlinear switching AMB system. The nonlinear observers are used to provide the estimate of the unmeasured state with an exponentially convergent error decay since the full state of the control system is not available for the feedback. The control systems of AMB setup is tested experimentally with base acceleration disturbance at different rotational speed. The bearable acceleration disturbance level of adaptive control was two times higher than PID control.
yl
xl
0
0.1
0.1
Displacement ( Z= 50 Hz )
0.5
Ŧ0.5 0
Ŧ0.04
VI. Conclusions
Control current ( Z= 50 Hz ) 5
i
i
5
[A]
Control current ( Z= 50 Hz )
0 Ŧ0.02
(a) (b) Fig.15 Acceleration disturbance
Fig.10 Orbit of the rotor(Adaptive Control) 5
0.02
Ŧ0.08
PID control case Time
0.5
Adaptive control case
Ŧ0.06
Ŧ0.08
Ŧ0.5 Ŧ0.5
Disturbance Signal ( Z=50 Hz ) 0.1
Adaptive control case
0.08
[V]
[V]
0.25
[ mm ]
0.25
yl Axis
[ mm ] yu Axis
Disturbance Signal ( Z=0 Hz )
Lower ( Z= 50 Hz ) 0.5
Amplitude
Upper ( Z= 50 Hz ) 0.5
fxl = (f5 − f7 ) fyl = (f6 − f8 )
(26)
The equations of motion obtained in (25) are derived according to the movement of the rotor’s center of mass. On the other hand, the measured signals are the displacements of the rotor at the lower and upper sensor locations. Since sensor locations are distinct from the mass center, the computation of the displacements of the rotor’s center of mass and angular displacements are necessary during control operation. Instead of computing the displacements xg , yg , θy and θx , the computation of the displacements of the rotor at the magnet locations makes the control system collocated. To this aim, the equations of the rotor AMB system may be transformed to the actuator locations. The displacement at the upper AMB location for the x direction is obtained as xu = xg + lu θy
0.5
(27)
Taking the double derivative of the above equation and substituting Equations (25) and (26) into the obtained deriva-
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tions, the equations are derived as
[2]
1 1 f1 − f3 x ¨u = x ¨g + lu θ¨y = M M 1 1 +lu f1 lu − f3 lu Ir Ir Ku i21 Ku i23 x ¨ u = au − (X0 − xu )2 (X0 + xu )2 where
au =
1 l2 + u M Ir
[3]
(28) [4]
[5]
(29)
[6]
The transformation of the equations for the location of lower AMB is realized with the same derivation procedure and follows as xl = xg − ll θy 1 1 f5 − f7 x¨l = x ¨g − ll θ¨y = M M 1 1 −ll − f5 ll + f7 ll I I r r 2 K l i5 Kl i27 x¨l = al − (X0 − xl )2 (X0 + xl )2 where
al =
1 l2 + l M Ir
(30)
(31)
Similarly, the displacement in the y direction is given as y u = y g − l u θx y l = y g + l l θx
(32)
Repeating the same procedure, the transformed equations are derived as Ku i22 Ku i24 − y¨u = au (Y0 − yu )2 (Y0 + yu )2 (33) Kl i26 Kl i28 y¨l = al − (Y0 − yl )2 (Y0 + yl )2 Note that in the above derivation process only the forces that directly effect the considered points are taken into account. When the transformation is done for the upper actuator location (xu , yu ) on the rotor, the upper AMB forces f1 and f3 are considered and f5 , f7 are taken as zero. For the lower actuator location (xl , yl ), the forces f5 and f7 are assumed to be effective and f1 , f3 are taken as zero.
Acknowledgment This work is supported by New Energy and Industrial Technology Development Organization(NEDO) of Japan with the Efficient Use Energy Program Grant Project. References [1]
F.J.M Thoolen, Development of an Advanced High Speed Flywheel Energy Storage System, Ph.D Thesis, Eindhoven University of Technology, 1993.
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Y. Ariga, K. Nonami, K. Sakai, Nonlinear Control of Zero Power Magnetic Bearing Using Lyapunov’s Direct Method, Proc. of the 7th ISMB, pp.293-298, 2000. S. Sivrioglu, K. Nonami, Adaptive Output Backstepping Control of a Flywheel Zero-Power AMB System with Parameter Uncertainty, Proc. of 42nd IEEE Conference on Decision and Control(CDC), pp.3942-3947, Hawaii-USA, 2003. S. Sivrioglu, K. Nonami, M. Saigo, Low Power Consumption Nonlinear Control with H-infinity Compensator for a ZeroBias Flywheel AMB System, Journal of Vibration and Control, vol.10, no. 8, pp.1151-1166, 2004. M.S. Queiroz, D.M. Dawson: Nonlinear Control of Active Magnetic Bearings: A Backstepping Appproach, IEEE Transactions on Control System Technology, Vol.4, No.5, pp. 545-552, 1996. M. Krstic, I. Kanellakopoulos, P. Kokotovic: Nonlinear and Adaptive Control Design, John Wiley& Sons, Inc., 1995.
